Elsevier

Wave Motion

Volume 70, April 2017, Pages 209-221
Wave Motion

Reflection and transmission of regular water waves by a thin, floating plate

https://doi.org/10.1016/j.wavemoti.2016.09.003Get rights and content

Highlights

  • Reflection and transmission of waves by thin plate.

  • Low reflection values when the plate is allowed to drift.

  • Wave transmission dependent by incident period and amplitude.

  • Relevant energy dissipation occurring in the presence of water overwash.

Abstract

Measurements of the wave fields reflected and transmitted by a thin floating plastic plate are reported for regular incident waves over a range of incident periods (producing wavelengths comparable to the plate length) and steepnesses (ranging from mild to storm-like). Two different plastics are tested, with different densities and mechanical properties, and three different configurations are tested. The configurations include freely floating plates, loosely moored plates (to restrict drift), and plates with edge barriers (to restrict waves overwashing the plates). The wave fields reflected and transmitted by plates without barriers are shown to become irregular, as the incident waves become steeper, particularly for the denser plastic and the moored plate. Further, the proportion of energy transmitted by the plates without barriers is shown to decrease as the incident wave becomes steeper, and this is related to wave energy dissipation.

Introduction

Thin, floating plates have been used to model sea ice floes (discrete chunks of sea ice) and very large floating structures (VLFSs, e.g. floating runways), with a large branch of these models developed to investigate interactions between surface-water waves and the plates (floes or VLFSs). For these applications, typically, the horizontal dimensions of the plates are comparable to wavelengths, so that the plates flex in response to the waves, in addition to experiencing rigid-body motions. Particularly for sea ice applications, wave–plate interaction models are used to predict the proportions of incident wave energy reflected and transmitted by the floe, as this provides predictions of the distances ocean waves travel into the ice-covered ocean and impact the ice cover  [1], [2], [3].

The canonical theoretical wave–plate interaction model is a Kirchoff–Love thin-plate floating on top of an inviscid, incompressible fluid undergoing irrotational motions, meaning the water velocity field can be defined as the gradient of a scalar potential function. It assumes linearity (in terms of the Bernoulli water pressure, the material response of the plate, and the moving boundary conditions) and harmonic time dependence at a prescribed angular frequency ω=2π/τ, thus fixing the open-water wavelength λ (for a given water depth). The plate oscillates in response to an incident wave at the prescribed frequency, in both its rigid-body and flexural modes, but does not drift. Water and plate motions are coupled at the lower surface of the plate only, assuming that all points on this surface remain in contact with the water during motion. This produces a boundary-value problem for the time-independent component of the velocity potential, ϕ, in which the plate cover provides a high-order surface condition, effectively removing the vertical geometry of the plate from the problem. Reflection and transmission result solely from impedance mismatches (i.e. different wave numbers) between the open water and the plate-covered water.

Meylan and Squire  [4] studied wave reflection and transmission by an ice floe of uniform thickness h, using a two-dimensional version of the canonical model (one horizontal dimension and one depth dimension, with coordinates x and z, respectively), similar to the theoretical model used in this study, although neglecting draught of the plate and its surge motion. Fig. 1 shows a schematic of the two-dimensional canonical model and the associated boundary-value problem for ϕ. The operator L involved in the surface condition is defined as L{}=ρg for the intervals of open water, where ρ is the water density and g9.81ms2 is the constant of gravitational acceleration. For the interval occupied by the plate, it is defined by L{}=(ρgω2m)+Eh3xxxx12(1ν2) where m is the plate mass per unit length, E is its Young’s modulus and ν is its Poisson’s ratio. Free edge conditions, ϕxxz=0 and ϕxxxz=0, are applied at the ends of the plate.

Motion is forced by a train of regular waves (with sinusoidal profiles), incident on the plate from its right-hand side, with surface elevation ηinc=acos(kx+ωt), where a is a prescribed amplitude and k=2π/λ is the open-water wave number. The plate partially reflects and partially transmits the incident waves. Far enough away from the plate that the exponentially decaying local motions have died out, the reflected and transmitted fields are regular wave trains, with surface elevations ηref=arefcos(kxωt+φref)andηtra=atracos(kx+ωt+φtra), respectively, where aref and atra are the reflected and transmitted amplitudes, and φref and φtra are phases. In the front field, on the right-hand side of the plate, the incident and reflected waves superpose to create a partial standing wave field. In the rear field, on the left-hand side, the wave field consists of transmitted waves only.

The canonical model is energy conserving, meaning that the energy in the incident waves is distributed into the reflected and transmitted waves. This property is expressed as R+T=1, where R=|aref/a|2 and T=|atra/a|2–the proportions of energy reflected and transmitted–are referred to as the reflection and transmission coefficients, respectively.

Meylan and Squire  [4] found that R is generally less than order 10−2 for plate lengths less than approximately one-third the incident wavelength, implying that the incident wave is almost entirely transmitted in this regime. Longer plates were found to reflect much greater proportions of the incident waves, with values of R typically order 10−1, although periodically vanishing as the plate length increased, due to resonance.

Montiel et al.  [5] showed that, for a plate with properties similar to those considered in this study although only half as thick, including the Archimedean draught of the plate in the model affects reflection for incident wavelengths approximately less than half the plate length only. Bennetts et al.  [6] and Smith and Meylan  [7] showed examples of the impacts of thickness variations on reflection and transmission.

Meylan and Squire  [8] used the canonical model with a circular disc to study wave scattering (reflection/transmission over all horizontal directions) by an ice floe, noting that  [9] earlier calculated scattering by a rigid circular floe. They found that, as the plate diameter increases with respect to the incident wavelength, the scattered energy increasingly focuses around the reflected and transmitted directions. Similarly, for elliptical plates, Bennetts and Williams  [10] found that directional scattering tends to pure reflection/transmission as the width of the plate increases.

Bennetts et al.  [11] used a laboratory experimental model, akin to the canonical theoretical model, to study transmission of regular incident waves by an ice floe. The experiments involved measurements of the surface elevation in the rear field of a loosely moored, square, plastic plate, with regular incident waves, for a range of plate thicknesses, incident wave periods and steepnesses, and two plastics. They showed that for incident steepnesses as small as ka=0.08, the plate could transmit highly irregular waves (in this case inducing a broad frequency content), with the irregularity increasing as the incident waves became steeper. They attributed this phenomenon to incident waves washing over the upper surface of the plate due to its small freeboard (referred to as overwash), then running off the plate into the rear field, producing high-frequency components in the transmitted wave spectrum. Overwash is a highly nonlinear phenomenon not included in the canonical model, but commonly noted in laboratory tests on wave interactions with thin floating plates/experimental models of wave–ice interactions (e.g.  [12], [13], [14], [15], [16]), and with  [17] reporting observations of seawater being washed onto the surfaces of Antarctic sea ice floes. Skene et al.  [18] analysed the depth of the overwash during  [11]’s experiments, using measurements provided by a wave gauge mounted on top of the plate, and developed an associated theoretical model, using predictions of plate motions and wave surface elevations surrounding the plate provided by the canonical model.

Bennetts and Williams  [19] studied transmission of regular incident waves through arrays of 40–80 loosely moored, circular, wooden plates during a series of laboratory wave-basin experiments. They showed that, for short incident periods, the proportion of wave energy transmitted drops sharply as the wave steepness increases, noting a correlation between this behaviour and the occurrence of overwash. Similarly, Meylan et al.  [20] used measurements of wave activity in the ice-covered Antarctic Ocean to provide evidence that the energy of short-period waves attenuate more rapidly with distance travelled as the incident steepness increases.

Toffoli et al.  [21] analysed transmission of regular incident waves by an unmoored plastic plate in a laboratory-experimental setting, replicating the two-dimensional version of the canonical model. They showed the theoretical model (including draught and surge) predicts transmitted amplitudes accurately for incident waves with steepness approximately ka0.06 only, and increasingly over predicts the transmitted amplitudes as the incident waves become steeper. Further, by extracting the reflected wave energy from surface-elevation measurements in the front field, they showed that the loss of model accuracy is correlated to wave-energy dissipation in the experiments, indicating wave breaking in the overwash as the likely sink.

An extended experimental dataset to that of Toffoli et al.  [21] is analysed in this investigation. The dataset includes tests on loosely moored plates (similarly to Bennetts et al.  [11], Bennetts and Williams  [19]), and plates with barriers around their edges to prevent overwash, respectively, to test the roles of drift and overwash on reflection and transmission. Moreover, it includes data from two different plastics, with different mechanical properties, and different densities (providing different freeboards and affecting the strength of overwash). The reflected wave field is explicitly analysed, in addition to the transmitted wave field. It is shown that the reflected and transmitted fields are regular for the plates with barriers, but become irregular as the incident steepness increases for the plates without barriers, with the irregularity stronger for the denser plate and when the plate is moored. The reflection coefficients produced by the unmoored plates are shown to be significantly smaller than those produced by the moored plates, for all but the longest-period incident waves tested. Consistent with the findings of Toffoli et al.  [21], the transmission coefficients are found to decrease with increasing incident steepness, but only for the plates without barriers. Evidence is provided to link the decrease in transmission coefficients with wave-energy dissipation.

Section snippets

Experimental setup and post processing

Experimental tests were conducted in the 60 m long and 2 m wide Extreme Air–Sea Interaction Flume, University of Melbourne, Australia. The flume is equipped with a cylinder-type wave-maker at one end, and a linear beach at the opposite end. The flume was filled with fresh water (density ρ1000kgm3) up to a depth of 0.9 m.

A 1 m long, 1.9 m wide and 0.01 m thick plastic sheet was placed in the flume, to act as the thin plate. Two different plastics were tested: (i) polypropylene (PP)1

The front field

Fig. 4 shows 25 s time windows of surface elevations in the front field produced by the PP plate for the three configurations, and for the mildest steepness, ka=0.06, and most storm-like steepness, ka=0.15. For the mild steepness, the front fields produced by both moored plates retain the regular profile of the incident field. The freely floating plate slowly drifts down the flume, so that the reflection source is moving away from the front gauges, producing a surface elevation in the form of a

Reflection and transmission coefficients and dissipation

Fig. 13 shows the reflection coefficient, R, as a function of the incident wave steepness, for both plastics and all three configurations. Error bars, equivalent to two times the standard deviation (i.e. 95% confidence intervals), show the sample variability. The data are slightly offset with respect to the incident steepness for the sake of clarity. Predictions given by a coupled potential-flow and thin-plate theoretical model are superimposed for reference (see §1, and  [21]), noting that the

Conclusions

Analysis of reflection and transmission of regular incident water waves by a thin floating plate during laboratory wave-flume experiments has been reported. A plastic sheet acted as the thin plate, and two plastics were tested (a more rigid and dense PP and a more flexible and less dense PVC). Three different deployment configurations were adopted: one in which the plate was moored and barriers were attached to its edge to suppress overwash; one in which the plate was moored without barriers,

Acknowledgements

The authors thank the two anonymous reviewers for their helpful comments. The Universities of Melbourne and Newcastle funded the experiments. FN is supported by a Swinburne University of Technology Postgraduate Research Award. The Australian Research Council funds an early-career fellowship for LGB (DE130101571) and a mid-career fellowship for JPM (FT120100409). The Australian Antarctic Science Program provides support for LGB and a Ph.D. top-up scholarship for DMS (project 4123). MHM

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